Mereotopology

Origin. Whitehead proposed region-based theories (1920s). Clarke formalized connection (1981). Combines mereology (parts) with topology (connection). Randell, Cui, Cohn: Region Connection Calculus (1992). Foundation for qualitative spatial reasoning without points.

Models. Regions with connection and parthood. Mereology: part-whole relations. Topology: open, closed, boundary, connection. Combined: regions that touch, overlap, contain. Points derived, not primitive. Intuitive for physical objects and space.

Formalism.

Primitives:

C(x, y): x connects with y (shares point or boundary)
Or: P(x, y): x is part of y

Connection-based (Clarke): C is reflexive and symmetric. x is connected to y iff they share a point (in any derived point topology).

Defined notions:

P(x,y) ≡ ∀z.(C(z,x) → C(z,y)) (parthood via connection)
O(x,y) ≡ ∃z.(P(z,x) ∧ P(z,y)) (overlap)
EC(x,y) ≡ C(x,y) ∧ ¬O(x,y) (external connection: touch but don't overlap)
DC(x,y) ≡ ¬C(x,y) (disconnected)

Interior and boundary:

IP(x,y): x is interior part of y (x inside y, not touching y's boundary)
TP(x,y): x is tangential part (part touching boundary)

RCC-8 relations (derived): DC, EC, PO, EQ, TPP, NTPP, TPPi, NTPPi (Disconnected, externally connected, partial overlap, equal, tangential proper part, non-tangential proper part, and inverses)

Axioms:

Connection is reflexive, symmetric
Supplementation: proper parts have complements
Closure under sum, product, complement (various strengths)

Symbols.

Symbol	Unicode	Name	Meaning
C	—	Connects	Shares boundary/point
P	—	Part	Parthood
O	—	Overlap	Share part
EC	—	External connection	Touch boundary
DC	—	Disconnected	No connection
PP	—	Proper part	Strict part
IP	—	Interior part	Inside
TP	—	Tangential part	Touching boundary

Metatheory. Various mereotopologies: weaker to stronger axioms. RCC-8 satisfiability is NP-complete. Path consistency tractable subclass. Models: regular closed sets in topological space. Connection definable from mereology + topology; parthood definable from connection. Categorical properties studied.

Applies to. Geographic information systems. Robot navigation. Natural language spatial expressions ("in," "touching," "near"). Architectural design. Medical imaging. Naive physics. Ontology (spatial entities).

Limitations. Qualitative only — no distances or coordinates. Different axiom systems yield different theories. Point-free but points often needed practically. 3D harder than 2D. Vague boundaries problematic. Integration with quantitative geometry complex. Tool support limited.